2. D10 is a boolean algebra example
1. `D_(10)` is a boolean algebra?
Solution:
(1) Commutative laws
The following are the tables for `+`,`*` and `'`
| `+` | 1 | 2 | 5 | 10 | | 1 | 1 `1=1+1=lcm(1,1)` | 2 `2=1+2=lcm(1,2)` | 5 `5=1+5=lcm(1,5)` | 10 `10=1+10=lcm(1,10)` | | 2 | 2 `2=2+1=lcm(2,1)` | 2 `2=2+2=lcm(2,2)` | 10 `10=2+5=lcm(2,5)` | 10 `10=2+10=lcm(2,10)` | | 5 | 5 `5=5+1=lcm(5,1)` | 10 `10=5+2=lcm(5,2)` | 5 `5=5+5=lcm(5,5)` | 10 `10=5+10=lcm(5,10)` | | 10 | 10 `10=10+1=lcm(10,1)` | 10 `10=10+2=lcm(10,2)` | 10 `10=10+5=lcm(10,5)` | 10 `10=10+10=lcm(10,10)` |
| | | `*` | 1 | 2 | 5 | 10 | | 1 | 1 `1=1*1=gcd(1,1)` | 1 `1=1*2=gcd(1,2)` | 1 `1=1*5=gcd(1,5)` | 1 `1=1*10=gcd(1,10)` | | 2 | 1 `1=2*1=gcd(2,1)` | 2 `2=2*2=gcd(2,2)` | 1 `1=2*5=gcd(2,5)` | 2 `2=2*10=gcd(2,10)` | | 5 | 1 `1=5*1=gcd(5,1)` | 1 `1=5*2=gcd(5,2)` | 5 `5=5*5=gcd(5,5)` | 5 `5=5*10=gcd(5,10)` | | 10 | 1 `1=10*1=gcd(10,1)` | 2 `2=10*2=gcd(10,2)` | 5 `5=10*5=gcd(10,5)` | 10 `10=10*10=gcd(10,10)` |
| | | `'` | 1 | 2 | 5 | 10 | | 10 `10=1'` | 5 `5=2'` | 2 `2=5'` | 1 `1=10'` |
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From the tables it is clear that `AA x, y in D_(10), x+yin D_(10) and x*yin D_(10)`
`:.` `+` and `*` are binary operations.
The symmetry about the principal diagonal of first two tables indicates the commutative laws hold.
(2) Associative laws
We verify associative laws for 1,2 and 5
`1+(2+5)=1+10=10`
`(1+2)+5=2+5=10`
and
`1*(2*5)=1*1=1`
`(1*2)*5=1*5=1`
Similarly, it may be verified for other cases.
(3) Distributive laws
We verify distributive laws for 2,5 and 10
`2+5*10=2+5=10`
`(2+5)*(2+10)=10*10=10`
and
`2*(5+10)=2*10=2`
`2*5+2*10=1+2=2`
Similarly, it may be verified for other cases.
(4) Existence of identity elements
For the zero and unit elements, we check
`1+1=1,1+2=2,1+5=5,1+10=10`
`10*1=1,10*2=2,10*5=5,10*10=10`
`:. 1` is the zero element and `10` is the unit element
(5) Existence of complement
`1+1'=1+10=10` & `1*1'=1*10=1`
`2+2'=2+5=10` & `2*2'=2*5=1`
`1'=10,``10'=1`
`2'=5,``5'=2`
`:. (D_(10),+,*,',1,10)` is a boolean algebra.
This material is intended as a summary. Use your textbook for detail explanation.
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